Classroom Resource

16.3 Negative Statements

Subject Area

Mathematics

Grade(s)

9, 10, 11, 12

Overview

This informational material will introduce negations and explain how to represent these logical statements with symbols in truth tables. It will introduce the term tautology and explain how a truth table can demonstrate a tautology. Practice questions with a PDF answer key are provided. In addition, there is a self-checking online practice tool.

Mathematics (2019) Grade(s): 09-12 - Applications of Finite Mathematics

MA19.FM.1

Represent logic statements in words, with symbols, and in truth tables, including conditional, biconditional, converse, inverse, contrapositive, and quantified statements.

Unpacked Content

UP:MA19.FM.1

Vocabulary

  • Proposition
  • Statement variables
  • Logical operators
  • Truth table
  • Negation
  • Conditional statement
  • Hypothesis/antecedent
  • Conclusion/consequent
  • Converse statement
  • Inverse statement
  • Contrapositive statement
  • Biconditional statement
  • Equivalent statements

Knowledge

Students know:

  • How to determine if a simple statement is true or false.

Skills

Students are able to:

  • Construct a truth table for propositions with a variety of operators.
  • Write a proposition using logical operators and statement variables such as p and q.
  • Write the converse, inverse, contrapositive and biconditional of a conditional statement using logical operators and statement variables.

Understanding

Students understand that:

  • A conditional statement’s validity is based on the validity of its components.
  • Truth tables must contain all possible assignments of true and false for each component.
  • A statement is either true or false.

Body

  1. Represent logic statements in words, with symbols, and in truth tables, including conditional, biconditional, converse, inverse, contrapositive, and quantified statements.
Mathematics (2019) Grade(s): 09-12 - Applications of Finite Mathematics

MA19.FM.2

Represent logic operations such as and, or, not, nor, and x or (exclusive or) in words, with symbols, and in truth tables.

Unpacked Content

UP:MA19.FM.2

Vocabulary

  • Compound statement
  • Negation
  • Conjunction
  • Disjunction

Knowledge

Students know:

  • A statement is either true or false.
  • A truth table must include every possible assignment of true and false for each component of a compound statement.

Skills

Students are able to:

  • Construct a truth table for a compound statement.
  • Represent compound statements using statement variables and logical operators.

Understanding

Students understand that:

  • The validity of the simple statements that make up a compound statement determine the compound statement’s validity.

Body

  1. Represent logic operations such as and, or, not, nor, and x or (exclusive or) in words, with symbols, and in truth tables.
Mathematics (2019) Grade(s): 09-12 - Applications of Finite Mathematics

MA19.FM.3

Use truth tables to solve application-based logic problems and determine the truth value of simple and compound statements including negations and implications.

Unpacked Content

UP:MA19.FM.3

Vocabulary

  • Equivalent statements or logical equivalence

Knowledge

Students know:

  • How to construct a truth table from a given logic statement.

Skills

Students are able to:

  • Represent an application-based logic problem as a statement(s) using logical operators and statement variables.
  • Construct a truth table to determine a solution to a logic problem.

Understanding

Students understand that:

  • Complex situations including logic problems can be modeled using truth tables.
  • Statements are logically equivalent if they have the same truth value for every possible assignment of true and false for each component.

Body

  1. Use truth tables to solve application-based logic problems and determine the truth value of simple and compound statements including negations and implications. a. Determine whether statements are equivalent and construct equivalent statements.
    Example: Show that the contrapositive of a statement is its logical equivalent.
Mathematics (2019) Grade(s): 09-12 - Applications of Finite Mathematics

MA19.FM.4

Determine whether a logical argument is valid or invalid, using laws of logic such as the law of syllogism and the law of detachment.

Unpacked Content

UP:MA19.FM.4

Vocabulary

  • Tautology
  • Contradiction
  • Law of syllogism
  • Law of detachment/modus ponens

Knowledge

Students know:

  • How to construct a truth table from a given logic statement.

Skills

Students are able to:

  • Construct valid arguments.
  • Identify the validity of arguments.

Understanding

Students understand that:

  • Truth tables can be used to construct a valid argument or to determine the validity of an argument.
  • In order for an argument to be valid, the form of the argument must be valid.

Body

  1. Determine whether a logical argument is valid or invalid, using laws of logic such as the law of syllogism and the law of detachment. a. Determine whether a logical argument is a tautology or a contradiction.
Link to Resource
https://flexbooks.ck12.org/cbook/ck-12-precalculus-concepts-2.0/section/16.3/pr…

CR Resource Type

Informational Material

Resource Provider

CK-12
Accessibility
License

License Type

Custom

Custom License Type (URL)

https://www.ck12info.org/terms-of-use/

Comments or Notes

Students can create a free account online to save their progress.

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